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1. Deformation Theory of Log Pseudo-holomorphic Curves and Logarithmic Ruan–Tian Perturbations

   https://doi.org/10.1007/s42543-023-00069-1  

   Farajzadeh-Tehrani, Mohammad (May 2023, Peking Mathematical Journal)

   In a previous paper (Farajzadeh-Tehrani in Geom Topol 26:989–1075, 2022), for any logarithmic symplectic pair (X, D) of a symplectic manifold X and a simple normal crossings symplectic divisor D, we introduced the notion of log pseudo-holomorphic curve and proved a compactness theorem for the moduli spaces of stable log curves. In this paper, we introduce a natural Fredholm setup for studying the deformation theory of log (and relative) curves. As a result, we obtain a logarithmic analog of the space of Ruan–Tian perturbations for these moduli spaces. For a generic compatible pair of an almost complex structure and a log perturbation term, we prove that the subspace of simple maps in each stratum is cut transversely. Such perturbations enable a geometric construction of Gromov–Witten type invariants for certain semi-positive pairs (X, D) in arbitrary genera. In future works, we will use local perturbations and a gluing theorem to construct log Gromov–Witten invariants of arbitrary such pair (X, D). 

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2. Relative vanishing theorems for Q-schemes

   https://doi.org/10.14231/AG-2025-003  

   Murayama, Takumi (December 2024, Algebraic Geometry)

   We prove the relative Grauert–Riemenschneider vanishing, Kawamata–Viehweg vanishing, and Kollár injectivity theorems for proper morphisms of schemes of equal characteristic zero, solving conjectures of Boutot and Kawakita. Our proof uses the Grothendieck limit theorem for sheaf cohomology and Zariski–Riemann spaces. We also show that these vanishing and injectivity theorems hold for locally Moishezon (respectively, projective) morphisms of quasi-excellent algebraic spaces and semianalytic germs of complex-analytic spaces (respectively, quasi-excellent formal schemes and non-Archimedean analytic spaces), all in equal characteristic zero. We give many applications of our vanishing results. For example, we extend Boutot’s theorem to all Noetherian Q-algebras by showing that pseudo-rationality descends under pure maps of Q-algebras. This solves a conjecture of Boutot and answers a question of Schoutens. The proofs of this Boutot-type result and of our vanishing and injectivity theorems all use a new characterization of rational singularities using Zariski–Riemann spaces. 

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3. A uniformization theorem for Stein spaces

   Huang, Xiaojun; Ming, Xiao (April 2020, Complex analysis and its synergies)

   In this paper, we establish the Qi-Keng Lu uniformization theorem for Stein spaces with possibly isolated normal singularities. 

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4. Embedding codimension of the space of arcs

   https://doi.org/10.1017/fmp.2021.19  

   Chiu, Christopher; de Fernex, Tommaso; Docampo, Roi (January 2022, Forum of Mathematics, Pi)

   Abstract We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg and Kazhdan on formal neighbourhoods in arc spaces by providing a converse to their theorem, an optimal bound for the embedding codimension of the formal model appearing in the statement, a precise formula for the embedding dimension of the model constructed in Drinfeld’s proof and a geometric meaningful way of realising the decomposition stated in the theorem. 

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5. The symplectic isotopy problem for rational cuspidal curves

   https://doi.org/10.1112/s0010437x2200762x  

   Golla, Marco; Starkston, Laura (July 2022, Compositio Mathematica)

   We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology. 

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